 Cardinality of integer-periodic function spaces

For fixed positive integer period $n$, integer-valued periodic functions on $\mathbb{Z}$ are in bijection with ${\mathbb{Z}}^n$ and hence form a countable set. Their union over periods is a countable union of countable sets. In contrast, period-one integer-valued functions on $\mathbb{Q}$ can prescribe arbitrary binary values on its infinitely many distinct classes modulo $\mathbb{Z}$. The power set of a countably infinite set injects into them, proving uncountability.